Experimentele Modale Analyse LES 1 THEORETISCHE INLEIDING Patrick

Overzicht cursus – – – Theoretische basisbegrippen Meettechnieken Niet-parametrische schattingen Parametrische schattingen Toepassingen –

3 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit

4 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit

5 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit

Frequency Response of MDOF System 6 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics &

Basic Equations of Motion for SDOF System – Time domain f(t) – Frequency domain

Poles of SDOF System l Poles : l l l Generalized eigenvalue problem 8

The SDOF Modal-Parameter Model Pole p – Re - Decay rate – Im -

Poles and Residues Amplitude of mode is not given by the residue alone Hi-Fi

Basic Equations for a 2 -DOF System Forces acting on mass 1 f 2(t)

Basic Equations for a 2 -DOF System Time domain Frequency domain (Laplace) Matrix notations

FRF of 2 -DOF System f 2(t) m 2 f 1(t) x 2(t) c

Multiple Degree of Freedom (MDOF) Dynamic stiffness matrix Transfer function matrix 14 EXPERIMENTELE MODALE

Poles of MDOF System Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION 15 EXPERIMENTELE MODALE

Eigenvalues and Eigenvectors No Damping 16 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics &

Weighted Orthogonality of Modal Vectors – Pre-multiply with – Transpose the equation 17 EXPERIMENTELE

Weighted Orthogonality of Modal Vectors (eq. 1) – – Substracting – – From eq.

Modal Mass and Stiffness No Damping – Modal mass and stiffness are not unique

Modal Coordinates 20 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group

Modal Model 21 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group

Modal Decomposition IRF FRF 22 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration

Scaling of the Mode Shapes 23 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics &

Proportional Damping – Eigenvalues 24 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration

Proportional Damping – Complex system poles and normal modal vectors – Modal mass, stiffness

General Viscous Damping 26 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research

General Viscous Damping – Eigenvalues and vectors – Complex system poles and complex modal

General Viscous Damping Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION 28 EXPERIMENTELE MODALE ANALYSIS,

General Viscous Damping 29 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research

Operational Deflection Shapes (ODS) Resonant frequencies (peaks) Mode shapes (ODS) 30 EXPERIMENTELE MODALE ANALYSIS,

Operational Deflection Shapes (ODS) Uncoupled modes (SDOF) 31 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005

Operational Deflection Shapes (ODS) Coupled modes + = 32 EXPERIMENTELE MODALE ANALYSIS, LES 1,

Force Appropriation Method – Also called: – Normal Mode Testing – Phase Resonance Testing

Force Appropriation Method – Real/imaginary part – Find (mono-phased) forced vector to obtain a

Mat. Lab Oefening modal_model. m – – SDOF MDOF Eigenwaarden en vectoren Schaling van

Slides: 35

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Experimentele Modale Analyse LES 1 – THEORETISCHE INLEIDING Patrick Guillaume E-mail: patrick. guillaume@vub. ac. be Tel. : 02/6293566 Faculty of Engineering Department of Mechanical Engineering ACOUSTICS & VIBRATION RESEARCH GROUP Pleinlaan 2 • B-1050 Brussel • Belgium avrg@vub. ac. be • http: //avrg. vub. ac. be 1 12/19/2021 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005

Overzicht cursus – – – Theoretische basisbegrippen Meettechnieken Niet-parametrische schattingen Parametrische schattingen Toepassingen – – – Gevoeligheidsanalyse FE Model “updating” “Substructuring” “Operationele” modale analyse “Structural Health Monitoring” “Flight Flutter Testing” 2 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

3 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

4 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

5 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Frequency Response of MDOF System 6 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Basic Equations of Motion for SDOF System – Time domain f(t) – Frequency domain (Laplace) x(t) – Dynamic Stiffness Z(s) (static) stiffness k – Transfer function – Frequency response function (FRF) 7 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Poles of SDOF System l Poles : l l l Generalized eigenvalue problem 8 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

The SDOF Modal-Parameter Model Pole p – Re - Decay rate – Im - Modal frequency Residue R – Strength of the mode 9 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Poles and Residues Amplitude of mode is not given by the residue alone Hi-Fi turntable Car – Same pole – Different residues 10 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Basic Equations for a 2 -DOF System Forces acting on mass 1 f 2(t) – m 2 f 1(t) x 2(t) c 2 k 2 m 1 k 1 – – x 1(t) c 1 – – – Sum forces = 0 11 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Basic Equations for a 2 -DOF System Time domain Frequency domain (Laplace) Matrix notations Dynamic stiffness matrix Transfer function matrix 12 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

FRF of 2 -DOF System f 2(t) m 2 f 1(t) x 2(t) c 2 k 2 m 1 k 1 x 1(t) c 1 0° -180° -360° 13 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Multiple Degree of Freedom (MDOF) Dynamic stiffness matrix Transfer function matrix 14 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Poles of MDOF System Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION 15 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Eigenvalues and Eigenvectors No Damping 16 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Weighted Orthogonality of Modal Vectors – Pre-multiply with – Transpose the equation 17 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Weighted Orthogonality of Modal Vectors (eq. 1) – – Substracting – – From eq. 1: 18 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Mass and Stiffness No Damping – Modal mass and stiffness are not unique ! 19 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Coordinates 20 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Model 21 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Modal Decomposition IRF FRF 22 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Scaling of the Mode Shapes 23 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Proportional Damping – Eigenvalues 24 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Proportional Damping – Complex system poles and normal modal vectors – Modal mass, stiffness and damping 25 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

General Viscous Damping 26 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

General Viscous Damping – Eigenvalues and vectors – Complex system poles and complex modal vectors – Orthogonality 27 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

General Viscous Damping Poles: FREQUENCY RESPONSE FUNCTION IMPULSE RESPONSE FUNCTION 28 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

General Viscous Damping 29 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Operational Deflection Shapes (ODS) Resonant frequencies (peaks) Mode shapes (ODS) 30 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Operational Deflection Shapes (ODS) Uncoupled modes (SDOF) 31 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Operational Deflection Shapes (ODS) Coupled modes + = 32 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Force Appropriation Method – Also called: – Normal Mode Testing – Phase Resonance Testing – Traditionally used for Ground Vibration Testing (GVT) of airplanes (large structures) – First method to use multiple inputs – Find (mono-phased) forced vector to obtain a (mono-phased) response vector with a 90 degree phase 33 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Force Appropriation Method – Real/imaginary part – Find (mono-phased) forced vector to obtain a (mono-phased) response vector with a 90 degree phase – Verification of normal mode tuning by turning off the excitations 34 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel

Mat. Lab Oefening modal_model. m – – SDOF MDOF Eigenwaarden en vectoren Schaling van de eigenvectoren 35 EXPERIMENTELE MODALE ANALYSIS, LES 1, 2005 Acoustics & Vibration Research Group Vrije Universiteit Brussel